Lemma 8.4.6. Let \mathcal{C} be a site. The (2, 1)-category of stacks over \mathcal{C} has 2-fibre products, and they are described as in Categories, Lemma 4.32.3.
Proof. Let f : \mathcal{X} \to \mathcal{S} and g : \mathcal{Y} \to \mathcal{S} be 1-morphisms of stacks over \mathcal{C} as defined above. The category \mathcal{X} \times _\mathcal {S} \mathcal{Y} described in Categories, Lemma 4.32.3 is a fibred category according to Categories, Lemma 4.33.10. (This is where we use that f and g preserve strongly cartesian morphisms.) It remains to show that the morphism presheaves are sheaves and that descent relative to coverings of \mathcal{C} is effective.
Recall that an object of \mathcal{X} \times _\mathcal {S} \mathcal{Y} is given by a quadruple (U, x, y, \phi ). It lies over the object U of \mathcal{C}. Next, let (U, x', y', \phi ') be second object lying over U. Recall that \phi : f(x) \to g(y), and \phi ' : f(x') \to g(y') are isomorphisms in the category \mathcal{S}_ U. Let us use these isomorphisms to identify z = f(x) = g(y) and z' = f(x') = g(y'). With this identifications it is clear that
as presheaves. However, as the fibred product in the category of presheaves preserves sheaves (Sites, Lemma 7.10.1) we see that this is a sheaf.
Let \mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I} be a covering of the site \mathcal{C}. Let (X_ i, \chi _{ij}) be a descent datum in \mathcal{X} \times _\mathcal {S} \mathcal{Y} relative to \mathcal{U}. Write X_ i = (U_ i, x_ i, y_ i, \phi _ i) as above. Write \chi _{ij} = (\varphi _{ij}, \psi _{ij}) as in the definition of the category \mathcal{X} \times _\mathcal {S} \mathcal{Y} (see Categories, Lemma 4.32.3). It is clear that (x_ i, \varphi _{ij}) is a descent datum in \mathcal{X} and that (y_ i, \psi _{ij}) is a descent datum in \mathcal{Y}. Since \mathcal{X} and \mathcal{Y} are stacks these descent data are effective. Thus we get x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U), and y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U) with x_ i = x|_{U_ i}, and y_ i = y|_{U_ i} compatibly with descent data. Set z = f(x) and z' = g(y) which are both objects of \mathcal{S}_ U. The morphisms \phi _ i are elements of \mathit{Isom}(z, z')(U_ i) with the property that \phi _ i|_{U_ i \times _ U U_ j} = \phi _ j|_{U_ i \times _ U U_ j}. Hence by the sheaf property of \mathit{Isom}(z, z') we obtain an isomorphism \phi : z = f(x) \to z' = g(y). We omit the verification that the canonical descent datum associated to the object (U, x, y, \phi ) of (\mathcal{X} \times _\mathcal {S} \mathcal{Y})_ U is isomorphic to the descent datum we started with. \square
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